PDPs for Hyperparameters.
Let \(S\subset \lbrace 1, 2, ..., d\rbrace \)  denote an index set of features, and let \(C = \lbrace 1, 2, ..., d\rbrace  \setminus S\)  be its complement.
The partial dependence (PD) function [1]} of \(c: \Lambda \rightarrow {R}\)  for hyperparameter(s) \(S\)  is defined asTo keep notation simple, we denote \(c(\lambda )\)  as a function of two arguments \((\lambda _S, \lambda _C)\)  to differentiate components in the index set \(S\)  from those in the complement. The integral shall be understood as a multiple integral of \(c\)  where \(\lambda _j\) , \(j \in C\) , are integrated out.
\(c_{S}(\lambda _S) := {E}_{\lambda _C}\left[c(\lambda )\right]=\int _{\Lambda _C} c(\lambda _S, \lambda _C)~\textrm {d}\mathbb {P}(\lambda _C).\) 
